Derivatives - Keep it on files

Function $f(x)$	Derivative $f'(x)$	Interval of differentiability
$$k$$	$$0$$	$$\mathbb{R}$$
$$x$$	$$1$$	$$\mathbb{R}$$
$$ax+b$$	$$a$$	$$\mathbb{R}$$
$$x^2$$	$$2x$$	$$\mathbb{R}$$
$$x^n$$	$$n x^{n-1}$$	$$\mathbb{R}$$
$$\frac{1}{x}=x^{-1}$$	$$-\frac{1}{x^2}$$	$$]-\infty;0[\text{ and }]0;+\infty[$$
$$\frac{1}{x^n}=x^{-n}$$	$$-\frac{n}{x^{n+1}}=-nx^{-n-1}$$	$$]-\infty;0[\text{ and }]0;+\infty[$$
$$\sqrt{x}$$	$$\frac{1}{2\sqrt{x}}=-nx^{-n-1}$$	$$]0;+\infty[$$
$$sin(x)$$	$$cos(x)$$	$$\mathbb{R}$$
$$cos(x)$$	$$-sin(x)$$	$$\mathbb{R}$$
$$tan(x)$$	$$1+tan^2(x)=\frac{1}{cos^2(x)}$$	$$]-\frac{\pi}{2};\frac{\pi}{2}[\text{ }\cup\text{ } ]\frac{\pi}{2} + k\ pi;\frac{\pi}{2} + (k+1)\pi[$$
$$ln(x)$$	$$\frac{1}{x}$$	$$]0;+\infty[$$
$$e^x$$	$$e^x$$	$$\mathbb{R}$$
$$arcsin(x)$$	$$\frac{1}{\sqrt{1-x^2}}$$	$$]-1;1[$$
$$arccos(x)$$	$$\frac{-1}{x^2+1}$$	$$]-1;1[$$
$$arctan(x)$$	$$\frac{-1}{\sqrt{1-x^2}}$$	$$\mathbb{R}$$
$$cosh(x)$$	$$sinh(x)$$	$$\mathbb{R}$$
$$sinh(x)$$	$$cosh(x)$$	$$\mathbb{R}$$
$$tanh(x)$$	$$1-tanh^2(x)=\frac{1}{cosh^2(x)}$$	$$\mathbb{R}$$
$$arcsinh(x)$$	$$\frac{1}{\sqrt{1+x^2}}$$	$$\mathbb{R}$$
$$arccosh(x)$$	$$\frac{1}{\sqrt{x^2-1}}$$	$$]1;+\infty[$$
$$arctanh(x)$$	$$\frac{1}{1-x^2}$$	$$]-1;1[$$
$$cotan(x)$$	$$\frac{-1}{sin^2(x)}$$	$$]0;+\pi[$$
$$arccotan(x)$$	$$\frac{-1}{x^2+1}$$	$$]-\infty;+\infty[$$
$$cotanh(x)$$	$$\frac{1}{sinh^2(x)}$$	$$\mathbb{R}$$
$$arccotanh(x)$$	$$\frac{1}{1-x^2}$$	$$\mathbb{R} \setminus \{-1;1 \}$$

Operation	Derivative
$$u+v$$	$$u’+v’$$
$$k.u \text{ (k a constant)}$$	$$k.u’$$
$$u.v$$	$$u’.v+v’.u$$
$$\frac{1}{u}$$	$$\frac{-u’}{u^2}$$
$$\frac{u}{v}$$	$$\frac{u’v-uv’}{v^2}$$
$$v \circ u$$	$$u'(v’ \circ u)$$
$$u^\alpha$$	$$\alpha u’.u^{\alpha-1}$$
$$\sqrt{u}$$	$$\frac{u’}{2\sqrt{u}}$$
$$cos(u)$$	$$-u’sin(u)$$
$$sin(u)$$	$$u’cos(u)$$
$$e^u$$	$$u’.e^u$$
$$ln(u)$$	$$\frac{u’}{u}$$
$$u(ax+b)$$	$$a.u'(ax+b)$$
$$(f.g)^n$$	$$\sum_{k=0}^n \left(\begin{array}{c} n \\ k \end{array}\right) f^k .g^{(n-k)}$$
$$(f^{-1})’$$	$$\frac{1}{f’ \circ f^{-1}}$$
$$\frac{1}{u^n}$$	$$\frac{-n.u’}{u^{n+1}}$$
$$tan(u)$$	$$u'(1+tan^2(u))=\frac{u’}{cos^2(u)}$$
$$arctan(u)$$	$$\frac{u’}{u^{2}+1}$$

Function \(f(x)\)	Derivative \(f'(x)\)	Interval of differentiability
$$k$$	$$0$$	$$\mathbb{R}$$
$$x$$	$$1$$	$$\mathbb{R}$$
$$ax+b$$	$$a$$	$$\mathbb{R}$$
$$x^2$$	$$2x$$	$$\mathbb{R}$$
$$x^n$$	$$n x^{n-1}$$	$$\mathbb{R}$$
$$\frac{1}{x}=x^{-1}$$	$$-\frac{1}{x^2}$$	$$]-\infty;0[\text{ and }]0;+\infty[$$
$$\frac{1}{x^n}=x^{-n}$$	$$-\frac{n}{x^{n+1}}=-nx^{-n-1}$$	$$]-\infty;0[\text{ and }]0;+\infty[$$
$$\sqrt{x}$$	$$\frac{1}{2\sqrt{x}}=-nx^{-n-1}$$	$$]0;+\infty[$$
$$sin(x)$$	$$cos(x)$$	$$\mathbb{R}$$
$$cos(x)$$	$$-sin(x)$$	$$\mathbb{R}$$
$$tan(x)$$	$$1+tan^2(x)=\frac{1}{cos^2(x)}$$	$$]-\frac{\pi}{2};\frac{\pi}{2}[\text{ }\cup\text{ } ]\frac{\pi}{2} + k\ pi;\frac{\pi}{2} + (k+1)\pi[$$
$$ln(x)$$	$$\frac{1}{x}$$	$$]0;+\infty[$$
$$e^x$$	$$e^x$$	$$\mathbb{R}$$
$$arcsin(x)$$	$$\frac{1}{\sqrt{1-x^2}}$$	$$]-1;1[$$
$$arccos(x)$$	$$\frac{-1}{x^2+1}$$	$$]-1;1[$$
$$arctan(x)$$	$$\frac{-1}{\sqrt{1-x^2}}$$	$$\mathbb{R}$$
$$cosh(x)$$	$$sinh(x)$$	$$\mathbb{R}$$
$$sinh(x)$$	$$cosh(x)$$	$$\mathbb{R}$$
$$tanh(x)$$	$$1-tanh^2(x)=\frac{1}{cosh^2(x)}$$	$$\mathbb{R}$$
$$arcsinh(x)$$	$$\frac{1}{\sqrt{1+x^2}}$$	$$\mathbb{R}$$
$$arccosh(x)$$	$$\frac{1}{\sqrt{x^2-1}}$$	$$]1;+\infty[$$
$$arctanh(x)$$	$$\frac{1}{1-x^2}$$	$$]-1;1[$$
$$cotan(x)$$	$$\frac{-1}{sin^2(x)}$$	$$]0;+\pi[$$
$$arccotan(x)$$	$$\frac{-1}{x^2+1}$$	$$]-\infty;+\infty[$$
$$cotanh(x)$$	$$\frac{1}{sinh^2(x)}$$	$$\mathbb{R}$$
$$arccotanh(x)$$	$$\frac{1}{1-x^2}$$	$$\mathbb{R} \setminus \{-1;1 \}$$

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